McGee will start main deconstruction and demolition works next month on the existing eight-storey building on the site which was formerly occupied by Cancer Research UK.
McGee will be on site until Spring 2018.
Nick Taylor, McGee’s Head of Demolition, said: “We have been fortunate enough to secure the project and are looking forward to working with LSE and their professional team.”
Julian Robinson, LSE’s Director of Estates said: “LSE has a very ambitious Estates Strategy but given our constrained location, doesn’t seem to have any easy projects.
“We need top quality contractors to deliver them and McGee has made an impressive start.”
The deconstruction and demolition of the existing building will make way for a new £140m scheme to be known as The Marshall Building.